reserve X for RealUnitarySpace;
reserve x, y, y1, y2 for Point of X;

theorem Th7:
  for S be Subset of X st S is summable_set
  for e be Real st
  0 < e ex Y0 be finite Subset of X st Y0 is non empty & Y0 c= S & for Y1 be
  finite Subset of X st Y0 c= Y1 & Y1 c= S holds |.((sum(S)).|.(sum(S))) - ((
  setsum(Y1)).|.(setsum(Y1))).| < e
proof
  let S be Subset of X such that
A1: S is summable_set;
  consider Y02 be finite Subset of X such that
  Y02 is non empty and
A2: Y02 c= S and
A3: for Y1 be finite Subset of X st Y02 c= Y1 & Y1 c= S holds ||.sum(S)
  - setsum(Y1).|| < 1 by A1,BHSP_6:def 3;
  let e be Real such that
A4: 0 < e;
  set e9 = e/(2*||.sum(S).||+1);
  0 <= ||.sum(S).|| by BHSP_1:28;
  then 0 <= 2*||.sum(S).||;
  then
A5: 0+0 < 2*||.sum(S).||+1;
  then 0 < e9 by A4,XREAL_1:139;
  then consider Y01 be finite Subset of X such that
A6: Y01 is non empty and
A7: Y01 c= S and
A8: for Y1 be finite Subset of X st Y01 c= Y1 & Y1 c= S holds ||.sum(S)
  - setsum(Y1).|| < e9 by A1,BHSP_6:def 3;
  set Y0 = Y01 \/ Y02;
A9: for Y1 be finite Subset of X st Y0 c= Y1 & Y1 c= S holds |.((sum(S))
  .|.(sum(S))) - ((setsum(Y1)).|.(setsum(Y1))).| < e
  proof
    let Y1 be finite Subset of X such that
A10: Y0 c= Y1 and
A11: Y1 c= S;
    set SS = sum(S)-setsum(Y1), SY = setsum(Y1);
    Y01 c= Y1 by A10,XBOOLE_1:11;
    then
A12: ||.SS.||*(2*||.sum(S).|| + 1) < e9*(2*||.sum(S).|| + 1) by A5,A8,A11,
XREAL_1:68;
    ||.SY.|| = ||.-SY.|| by BHSP_1:31
      .= ||.0.X - SY.|| by RLVECT_1:14
      .= ||.-sum(S) + sum(S) - SY.|| by RLVECT_1:5
      .= ||.-sum(S) + SS.|| by RLVECT_1:def 3;
    then ||.SY.|| <= ||.-sum(S).|| + ||.SS.|| by BHSP_1:30;
    then
A13: ||.SY.|| <= ||.sum(S).|| + ||.SS.|| by BHSP_1:31;
    Y02 c= Y1 by A10,XBOOLE_1:11;
    then ||.SS.|| + ||.SY.|| < 1 + (||.sum(S).|| + ||.SS.||) by A3,A11,A13,
XREAL_1:8;
    then ||.SY.|| + ||.SS.|| - ||.SS.|| < 1 + ||.sum(S).|| + ||.SS.|| - ||.SS
    .|| by XREAL_1:14;
    then
A14: ||.sum(S).|| + ||.SY.|| < 1 + ||.sum(S).|| + ||.sum(S).|| by XREAL_1:8;
    0 <= ||.SS.|| by BHSP_1:28;
    then ||.SS.||*(||.sum(S).|| + ||.SY.||) <= ||.SS.||*(2*||.sum(S).|| + 1)
    by A14,XREAL_1:64;
    then ||.SS.||*(||.sum(S).|| + ||.SY.||) + ||.SS.||*(2*||.sum(S).|| + 1) <
    e9*(2*||.sum(S).|| + 1) + ||.SS.||*(2*||.sum(S).|| + 1) by A12,XREAL_1:8;
    then ||.SS.||*(||.sum(S).|| + ||.SY.||) + ||.SS.||*(2*||.sum(S).|| + 1) -
||.SS.||*(2*||.sum(S).|| + 1) < e9*(2*||.sum(S).|| + 1) + ||.SS.||*(2*||.sum(S)
    .|| + 1) - ||.SS.||*(2*||.sum(S).|| + 1) by XREAL_1:14;
    then
A15: ||.SS.||*(||.sum(S).|| + ||.SY.||) < e by A5,XCMPLX_1:87;
    set F = (sum(S)).|.(sum(S)), G = (setsum(Y1)).|.(setsum(Y1));
    |.F - G.| = |.F - ((sum(S)).|.SY) + (((sum(S)).|.SY) - G).|
      .= |.((sum(S)).|.SS) + (((sum(S)).|.SY) - G).| by BHSP_1:12
      .= |.((sum(S)).|.SS) + (SS.|.SY).| by BHSP_1:11;
    then
A16: |.F - G.| <= |.((sum(S)).|.SS).| + |.SS.|.SY.| by COMPLEX1:56;
    |.((sum(S)).|.SS).| <= ||.sum(S).||*||.SS.|| by BHSP_1:29;
    then
    |.F - G.| + |.((sum(S)).|.SS).| <= |.((sum(S)).|.SS).| + |.SS.|.
    SY.| + ||.sum(S).||*||.SS.|| by A16,XREAL_1:7;
    then
    |.F - G.| + |.((sum(S)).|.SS).| <= (|.SS.|.SY.| + ||.sum(S).||*||.
    SS.||) + |.((sum(S)).|.SS).|;
    then
A17: |.F - G.| <= |.SS.|.SY.| + ||.sum(S).||*||.SS.|| by XREAL_1:6;
    |.SS.|.SY.| <= ||.SS.||*||.SY.|| by BHSP_1:29;
    then |.F - G.| + |.SS.|.SY.| <= |.SS.|.SY.| + ||.sum(S).||*||.SS.|| +
    ||.SS.||*||.SY.|| by A17,XREAL_1:7;
    then |.F - G.| + |.SS.|.SY.| <= ||.sum(S).||*||.SS.|| + ||.SS.||*||.SY
    .|| + |.SS.|.SY.|;
    then |.F - G.| <= ||.SS.||*||.sum(S).|| + ||.SS.||*||.SY.|| by XREAL_1:6;
    then |.F - G.| + ||.SS.||*(||.sum(S).|| + ||.SY.||) < e + ||.SS.||*(||.
    sum(S).|| + ||.SY.||) by A15,XREAL_1:8;
    then
    |.F - G.| + ||.SS.||*(||.sum(S).|| + ||.SY.||) - ||.SS.||*(||.sum(S)
.|| + ||.SY.||) < e + ||.SS.||*(||.sum(S).|| + ||.SY.||) - ||.SS.||*(||.sum(S)
    .|| + ||.SY.||) by XREAL_1:14;
    hence thesis;
  end;
  Y0 c= S by A7,A2,XBOOLE_1:8;
  hence thesis by A6,A9;
end;
